The fundamental theorem of calculus is appropriately named because it establishes a connection between differentiation and integration. Differentiation arose from the tangent problem, whereas integration arose from the area problem. Newton's teacher at Cambridge, Isaac Barrow, discovered that these two problems are actually closely related. In fact, he realized that differentiation and integration are inverse processes. The fundamental theorem of calculus gives the precise inverse relationship between the derivative and the integral. It was Newton and Leibniz who exploited this relationship and used it to develop calculus into a systematic mathematical model.

The fundamental theorem of calculus comes in two parts. The first part says that the definite integral of a continuous function is a differentiable function of its upper limit of integration. The second part says that the definite integral of a continuous function from a to b can be found from any one of the function's anti-derivatives Φ as the number Φ(b) – Φ(a).

The fundamental theorem of calculus, part 1: If the function f(x) is continuous on a closed interval [a, b], then the function g(x) defined by g(x) = where a ≤ x ≤ b is continuous on [a, b] and differentiable on (a, b) and g'(x) = f(x).

Ex: Use part 1 of the fundamental theorem of calculus to find the derivatives of the functions: (i) g(x) = , (ii) g(x) = and (iii) g(y) = .

Sol: (i) Since f(t) = is continuous, part 1 of the fundamental theorem of calculus gives g'(x) = ; (ii) We know that = . Since f(t) = –Cos (t2) is continuous, part 1 of the fundamental theorem of calculus gives g'(x) = –Cos (x2); (iii) Since f(t) = Sin t is continuous, part 1 of the fundamental theorem of calculus gives g'(y) = Sin y.

The fundamental theorem of calculus, part 2: If the function f(x) is continuous on a closed interval [a, b] and if Φ(x) is the anti-derivative of f(x) i.e, [Φ(x)] = f(x) then